Hurdles for recent measures in eternal inflation

In recent literature on eternal inflation, a number of measures have been introduced which attempt to assign probabilities to different pocket universes by counting the number of each type of pocket according to a specific procedure. We give an overview of the existing measures, pointing out some interesting connections and generic predictions. For example, pairs of vacua that undergo fast transitions between themselves will be strongly favored. The resultant implications for making predictions in a generic potential landscape are discussed. We also raise a number of issues concerning the types of transitions that observers in eternal inflation are able to experience.

Figure 1

A summary of the connections between the various measures. Solid green lines indicate equivalence between the measures for a terminal landscape. Dashed blue lines indicate equivalence in the case of a fully recycling landscape. Dashed-dotted red lines indicate that the measures assign the same relative weights to terminal vacua.

Figure 2

Some sample landscapes. Potential $V_1$ depicts the $ABZ$ example discussed by Bousso [1]. $V_2$ splits the $B$ vacuum by introducing a small barrier. The potential $V_3$ lowers the $A$ vacuum to zero or negative energy, so that it becomes terminal. The potential $V_4$ has a low-energy minimum with high-energy neighbors that have short lifetimes (relative to other vacua in the landscape).

Figure 3

A picture of an eternally inflating universe which takes into account both L and R tunneling geometries. At the bottom, there is an “original” parcel of comoving volume (defined by the horizontal spacelike slice at the bottom of the figure), which evolves in time (vertically). True and false vacuum bubble nucleation events occur via the R geometry in this volume, denoted by the shaded regions which in the case of true vacuum bubbles grow to a comoving Hubble volume and in the case of false vacuum bubbles shrink to a comoving Hubble volume. The vertical black lines denote the black holes formed during L-geometry tunneling events. On the other side of a wormhole (inside the captions), the initial distribution, which is fixed by the tunneling geometry, undergoes L and R tunneling events as well, spawning more disconnected parcels of volume in which this process repeats. The original parcel of comoving volume will spawn an infinite amount of new comoving volume via L-geometry tunneling events. Shown on the bottom of each parcel is the set of bubble shadows that might be used in the CHC method to calculate probabilities $P^{{\mathcal{V}}_i}$ for each region ${\mathcal{V}}_i$.

Figure 4

Examples of “snowman diagrams” summarizing relative transition probabilities ${\mu }_{NM}$. The one on the left is for a recycling landscape and the one on the right is for a terminal landscape.